Computer Graphics
- Modeling Basics -
Hanyang University
Jong-Il Park
Objectives
 Introduce simple data structures for building
polygonal models


Vertex lists
Edge lists
 OpenGL vertex arrays
Division of Electrical and Computer Engineering, Hanyang University
Representing a Mesh
 Consider a mesh
e2
v6
e8
e1
e6
e3
e9
v8
e11
v1
v5
e7
v7
v2
e12
v4
e10
e5
e4
v3
 There are 8 nodes and 12 edges


5 interior polygons
6 interior (shared) edges
 Each vertex has a location vi = (xi yi zi)
Division of Electrical and Computer Engineering, Hanyang University
Simple Representation
 Define each polygon by the geometric locations
of its vertices
 Leads to OpenGL code such as
glBegin(GL_POLYGON);
glVertex3f(x1, x1, x1);
glVertex3f(x6, x6, x6);
glVertex3f(x7, x7, x7);
glEnd();
 Inefficient and unstructured


Consider moving a vertex to a new location
Must search for all occurrences
Division of Electrical and Computer Engineering, Hanyang University
Inward and Outward Facing
Polygons
 The order {v1, v6, v7} and {v6, v7, v1} are equivalent in
that the same polygon will be rendered by OpenGL
but the order {v1, v7, v6} is different
 The first two describe outwardly facing polygons
 Use the right-hand rule =
counter-clockwise encirclement
of outward-pointing normal
 OpenGL can treat inward and
outward facing polygons differently
Division of Electrical and Computer Engineering, Hanyang University
Geometry vs Topology
 Generally it is a good idea to look for data structures
that separate the geometry from the topology




Geometry: locations of the vertices
Topology: organization of the vertices and edges
Example: a polygon is an ordered list of vertices with
an edge connecting successive pairs of vertices and
the last to the first
Topology holds even if geometry changes
Division of Electrical and Computer Engineering, Hanyang University
Vertex Lists
 Put the geometry in an array
 Use pointers from the vertices into this array
 Introduce a polygon list
P1
P2
P3
P4
P5
topology
v1
v7
v6
v8
v5
v6
geometry
x1 y 1 z1
x2 y 2 z2
x3 y 3 z3
x4 y 4 z4
x5 y5 z5.
x6 y 6 z6
x7 y 7 z7
x8 y 8 z8
Division of Electrical and Computer Engineering, Hanyang University
Shared Edges
 Vertex lists will draw filled polygons correctly
but if we draw the polygon by its edges, shared
edges are drawn twice
 Can store mesh by edge list
Division of Electrical and Computer Engineering, Hanyang University
Edge List
e2
e1
e2
e3
e4
e5
e6
e7
e8
e9
v1
v6
x1 y1 z1
x2 y2 z2
x3 y3 z3
x4 y4 z4
x5 y5 z5.
x6 y6 z6
x7 y7 z7
x8 y8 z8
v6
e8
e1
e6
e3
e9
v8
e11
v1
v5
e7
v7
v2
e12
e10
e5
e4
v3
Note polygons are
not represented
Division of Electrical and Computer Engineering, Hanyang University
Modeling a Cube
Model a color cube for rotating cube program
Define global arrays for vertices and colors
GLfloat vertices[][3] =
{{-1.0,-1.0,-1.0},{1.0,-1.0,-1.0},
{1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0},
{1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}};
GLfloat colors[][3] = {{0.0,0.0,0.0},{1.0,0.0,0.0},
{1.0,1.0,0.0}, {0.0,1.0,0.0}, {0.0,0.0,1.0},
{1.0,0.0,1.0}, {1.0,1.0,1.0}, {0.0,1.0,1.0}};
Division of Electrical and Computer Engineering, Hanyang University
Drawing a polygon from a list of
indices
Draw a quadrilateral from a list of indices into the
array vertices and use color corresponding to
first index
void polygon(int a, int b, int c , int d)
{
glBegin(GL_POLYGON);
glColor3fv(colors[a]);
glVertex3fv(vertices[a]);
glVertex3fv(vertices[b]);
glVertex3fv(vertices[c]);
glVertex3fv(vertices[d]);
glEnd();
}
Division of Electrical and Computer Engineering, Hanyang University
Draw cube from faces
void colorcube( )
{
polygon(0,3,2,1);
polygon(2,3,7,6);
polygon(0,4,7,3);
polygon(1,2,6,5);
polygon(4,5,6,7);
polygon(0,1,5,4);
}
5
6
2
1
7
4
0
3
Note that vertices are ordered so that
we obtain correct outward facing normals
Division of Electrical and Computer Engineering, Hanyang University
Efficiency
 The weakness of our approach is that we are building
the model in the application and must do many
function calls to draw the cube
 Drawing a cube by its faces in the most straight
forward way requires




6 glBegin, 6 glEnd
6 glColor
24 glVertex
More if we use texture and lighting
Division of Electrical and Computer Engineering, Hanyang University
Vertex Arrays
 OpenGL provides a facility called vertex arrays
that allows us to store array data in the
implementation
 Six types of arrays supported






Vertices
Colors
Color indices
Normals
Texture coordinates
Edge flags
 We will need only colors and vertices
Division of Electrical and Computer Engineering, Hanyang University
Initialization
 Using the same color and vertex data, first we
enable
glEnableClientState(GL_COLOR_ARRAY);
glEnableClientState(GL_VERTEX_ARRAY);
 Identify location of arrays
glVertexPointer(3, GL_FLOAT, 0, vertices);
data array
3d arrays
stored as floats
data contiguous
glColorPointer(3, GL_FLOAT, 0, colors);
Division of Electrical and Computer Engineering, Hanyang University
Mapping indices to faces
 Form an array of face indices
GLubyte cubeIndices[24] = {0,3,2,1,2,3,7,6
0,4,7,3,1,2,6,5,4,5,6,7,0,1,5,4};
 Each successive four indices describe a face of the
cube
 Draw through glDrawElements which replaces
all glVertex and glColor calls in the display
callback
Division of Electrical and Computer Engineering, Hanyang University
Drawing the cube
 Method 1:
what to draw
number of indices
for(i=0; i<6; i++) glDrawElements(GL_POLYGON, 4,
GL_UNSIGNED_BYTE, &cubeIndices[4*i]);
format of index data
start of index data
 Method 2:
glDrawElements(GL_QUADS, 24,
GL_UNSIGNED_BYTE, cubeIndices);
Draws cube with 1 function call!!
Division of Electrical and Computer Engineering, Hanyang University